What is a good introduction to integrable models in physics?

Question

I would be interested in a good mathematician-friendly introduction to (exactly solvable)  integrable models in physics, either a book or expository article.

Related MathOverflow question: what-is-an-integrable-system.

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user Phira

Comments

Do you have anything specific in mind? I think the term integrability is sometimes used in slightly different contexts.

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user Pieter Naaijkens

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The fact that "integrability" can mean so many things sometimes makes the quest to learn about it so challenging! I have found the introductory sections of Etingoff's paper www-math.mit.edu/~etingof/zlecnew.pdf to be a very good mathematical reference for a particular, physically interesting system (Calogero-Moser) which describes particles interacting in one-dimension.

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user Eric Zaslow

Answer 1

I take "integrable models" to mean "exactly solvable models in statistical physics".

You can take a look at the classic book

• R. J. Baxter - Exactly Solved Models in Statistical mechanics (You can download it for free)

Otherwise this new book is quit readable and covers more than just solvable models

• G. Mussardo - Statistical field theory: an introduction to exactly solved models in statistical physics

Others can probably give you more mathematician-friendly references, but I think it would be good if you could be more specific about what you are looking for.

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user Heidar

Comments

Yes, "exactly solvable" is what I mean. Thanks, I will update my question later.

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user Phira

Answer 2

This is a late response. Here are some more

• The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension

• Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems

• Exactly Solved Models: A Journey in Statistical Mechanics : Selected Papers with Commentaries

• Introduction to Classical Integrable Systems

• Classical Many-Body Problems Amenable to Exact Treatments

• Exact Methods in Low-dimensional Statistical Physics and Quantum Computing

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user Vijay Murthy

Answer 3

Another good recent book:

• Maciej Dunajski, Solitons, Instantons and Twistors

This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user just-learning

Comments

Also see his lecture notes on integrability.

Answer 4

These lecture notes by Babelon could be quite helpful.

Comments

There is a much more detailed ''Introduction to classical integrable systems'' by O. Babelon, D. Bernard, and M. Talon, very much recommended!

Answer 5

The slides from the lectures on integrable systems by Alexei Bolsinov from this course could be quite helpful (to get to the slides click on the Files tab at the above link). Also see the lecture notes by Boris Dubrovin "Integrable Systems and Riemann Surfaces".

Answer 6

Specifically for integrable partial differential systems which are dispersionless a.k.a. hydrodynamic-type, i.e. can be written as quasilinear first-order homogeneous systems, in the case of two independent variables see these lecture notes, Section 3 of this article, and references therein; in the case of threeindependent variables see Section 3 of this same article,  subsubsection 10.3.3 of the Dunajski book mentioned by @just-learning, and these lecture notes, and references therein; for the case of four independent variables see introductory part of this article, the Dunajski book again, and references therein.